Minors and cofactors

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The minor $M_{ij}(\mathbf{A})$ of an n-by-n square matrix $\mathbf{A}$ is the determinant of a smaller square matrix obtained by removing the row $i$ and the column $j$ from $\mathbf{A}$.

Multiplying the minor with $(-1)^{i+j}$ results in the cofactor $C_{ij}(\mathbf{A})$:

$C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})$

Example: Minors and cofactors of a 3-by-3 matrix $\mathbf{A}_3 = \left[\begin{array}{ccc} 1&0&1\\ 3&1&0\\ 1&0&2 \end{array}\right]$ The minors $M_{22}(\mathbf{A}_3)$ and $M_{31}(\mathbf{A}_3)$ for example are defined as $M_{22}(\mathbf{A}_3)= \left|\begin{array}{ccc} 1&\Box&1\\ \Box&\Box&\Box\\ 1&\Box&2 \end{array}\right|= \left|\begin{array}{cc} 1&1\\ 1&2 \end{array}\right|=2-1=1$ $M_{31}(\mathbf{A}_3)= \left|\begin{array}{ccc} \Box & 0&1\\ \Box & 1&0\\ \Box & \Box & \Box\\ \end{array}\right|= \left|\begin{array}{cc} 0&1\\ 1&0 \end{array}\right|=0-1=-1$ The corresponding cofactors in that case are $C_{22}(\mathbf{A}_3)=(-1)^{2+2}M_{22}(\mathbf{A}_e)=(-1)^4\cdot1=1$ $C_{31}(\mathbf{A}_3)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1$

Example: Minors and cofactors of a 4-by-4 matrix This example uses the transformation matrix $^R\mathbf{T}_N$ that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages. $^R\mathbf{T}_N = \left[\begin{array}{cccc} 0 & 1 & 0 & 2a\\ 0 & 0 & -1 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]$ The minors $M_{31}(^R\mathbf{T}_N)$ and $M_{42}(^R\mathbf{T}_N)$ for example are defined as $M_{31}(^R\mathbf{T}_N)= \left|\begin{array}{cccc} \Box & 1 & 0 & 2a\\ \Box & 0 & -1 & 0\\ \Box & \Box & \Box & \Box\\ \Box & 0 & 0 & 1 \end{array}\right|= \left|\begin{array}{ccc} 1&0&2a\\ 0&-1&0\\ 0&0&1 \end{array}\right|=-1-0=-1$ $M_{42}(^R\mathbf{T}_N)= \left|\begin{array}{cccc} 0&\Box&0&2a\\ 0&\Box&-1&0\\ -1&\Box&0&0\\ \Box & \Box & \Box & \Box\\ \end{array}\right|= \left|\begin{array}{ccc} 0&0&2a\\ 0&-1&0\\ -1&0&0 \end{array}\right|=0-2a=-2a$ The corresponding cofactors in that case are $C_{31}(^R\mathbf{T}_N)=(-1)^{3+1}M_{31}(^R\mathbf{T}_N)=(-1)^4\cdot(-1)=-1$ $C_{42}(^R\mathbf{T}_N)=(-1)^{4+2}M_{42}(^R\mathbf{T}_N)=(-1)^6\cdot(-2a)=-2a$